Solutions of Question 16 and 17 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.
How many odd numbers can be formed by using the digits $1,2,3,4,5,6$ when repetition of digit is not allowed?
Solution.
If digit at units place is odd, number is odd.
We shall fix unit place with $1, 3$ or $5$
and calculate arrangements of remaining digits.
Number of six digit odd numbers $=3 \times{ }^{5} P_{5}=360$
How many $4$-digit odd numbers can be formed using the digits $1,2,3,4$ and $5$ if no digit is repeated?
Solution.
Four digit no. is odd if unit place have an odd digit.
If at units place we put $1$
than possible arrangements for remaining three places are $={ }^{4} P_{3}=24$
When we place $3$ on units place,
we have again $24$ arrangements and
similarly for $5$ at units place there are $24$ arrangements.
Hence in total, we have $24+24+24=72$ possible $4$ digits odd numbers.